I am trying to download a kml file from Wikimapia containing information on quaters (districts), the very same information that is nicely available on Wikimapia. What I want to get at the end of the day is a shapefile containing as many polygons as there are districts in Milan to use in my GIS.

From this I get a tiny kml.part file, that I then rename .kml. The file doesn't open in Google Earth or QGIS. If I open it with a text editor I can just few lines of xml, but no information concerning my geometries...

There is also an example of how your result file is suppose to look like. Something I'm not sure about is the category. Maybe you should check the complete list so you can use an ID or try it without using this parameter to see if you get a valid result then.